At early times, before the CMB era, some process produces As discussed in Ref. [1]}, as we will assume here, this process can occur just after Big Bang nucleosynthesis (BBN) the WZDR \(\psi \)  and \(\phi \)  particles with an energy density equivalent to \(N_\text{UV}\)  additional neutrino species.
As \(\phi \)  and \(\psi \)  always maintain chemical and kinetic equilibrium,
once the temperature of this sector decreases below \(m_\phi \) , the scalars decay and annihilate,
depositing their entropy into the lighter \(\psi \)  species.
Due to this process which takes approximately a decade in redshift, the relative energy density of the fluid,
as quantified by the effective number of additional neutrinos species \(N (z)\) , increases to a value \(N_\text{IR}= (15/7)^{1/3} N_\text{UV}\) .
Assuming a temperature of the dark radiation today of \(T_{d0}\) , the transition approximately starts at redshift \(1+z_t = m_\phi / T_{d0}\) ,
and the evolution of \(N(z)\)  as a function of redshift is calculated by solving the entropy conservation equation.
